Step |
Hyp |
Ref |
Expression |
1 |
|
infxpidm2 |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝐴 × 𝐴 ) ≈ 𝐴 ) |
2 |
|
infn0 |
⊢ ( ω ≼ 𝐴 → 𝐴 ≠ ∅ ) |
3 |
2
|
adantl |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → 𝐴 ≠ ∅ ) |
4 |
|
fseqen |
⊢ ( ( ( 𝐴 × 𝐴 ) ≈ 𝐴 ∧ 𝐴 ≠ ∅ ) → ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ≈ ( ω × 𝐴 ) ) |
5 |
1 3 4
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ≈ ( ω × 𝐴 ) ) |
6 |
|
xpdom1g |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( ω × 𝐴 ) ≼ ( 𝐴 × 𝐴 ) ) |
7 |
|
domentr |
⊢ ( ( ( ω × 𝐴 ) ≼ ( 𝐴 × 𝐴 ) ∧ ( 𝐴 × 𝐴 ) ≈ 𝐴 ) → ( ω × 𝐴 ) ≼ 𝐴 ) |
8 |
6 1 7
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( ω × 𝐴 ) ≼ 𝐴 ) |
9 |
|
endomtr |
⊢ ( ( ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ≈ ( ω × 𝐴 ) ∧ ( ω × 𝐴 ) ≼ 𝐴 ) → ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ≼ 𝐴 ) |
10 |
5 8 9
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ≼ 𝐴 ) |
11 |
|
numdom |
⊢ ( ( 𝐴 ∈ dom card ∧ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ≼ 𝐴 ) → ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∈ dom card ) |
12 |
10 11
|
syldan |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∈ dom card ) |
13 |
|
eliun |
⊢ ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↔ ∃ 𝑛 ∈ ω 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) |
14 |
|
elmapi |
⊢ ( 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) → 𝑥 : 𝑛 ⟶ 𝐴 ) |
15 |
14
|
ad2antll |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝑛 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) ) → 𝑥 : 𝑛 ⟶ 𝐴 ) |
16 |
15
|
frnd |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝑛 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) ) → ran 𝑥 ⊆ 𝐴 ) |
17 |
|
vex |
⊢ 𝑥 ∈ V |
18 |
17
|
rnex |
⊢ ran 𝑥 ∈ V |
19 |
18
|
elpw |
⊢ ( ran 𝑥 ∈ 𝒫 𝐴 ↔ ran 𝑥 ⊆ 𝐴 ) |
20 |
16 19
|
sylibr |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝑛 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) ) → ran 𝑥 ∈ 𝒫 𝐴 ) |
21 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝑛 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) ) → 𝑛 ∈ ω ) |
22 |
|
ssid |
⊢ 𝑛 ⊆ 𝑛 |
23 |
|
ssnnfi |
⊢ ( ( 𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑛 ) → 𝑛 ∈ Fin ) |
24 |
21 22 23
|
sylancl |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝑛 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) ) → 𝑛 ∈ Fin ) |
25 |
|
ffn |
⊢ ( 𝑥 : 𝑛 ⟶ 𝐴 → 𝑥 Fn 𝑛 ) |
26 |
|
dffn4 |
⊢ ( 𝑥 Fn 𝑛 ↔ 𝑥 : 𝑛 –onto→ ran 𝑥 ) |
27 |
25 26
|
sylib |
⊢ ( 𝑥 : 𝑛 ⟶ 𝐴 → 𝑥 : 𝑛 –onto→ ran 𝑥 ) |
28 |
15 27
|
syl |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝑛 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) ) → 𝑥 : 𝑛 –onto→ ran 𝑥 ) |
29 |
|
fofi |
⊢ ( ( 𝑛 ∈ Fin ∧ 𝑥 : 𝑛 –onto→ ran 𝑥 ) → ran 𝑥 ∈ Fin ) |
30 |
24 28 29
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝑛 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) ) → ran 𝑥 ∈ Fin ) |
31 |
20 30
|
elind |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ ( 𝑛 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) ) → ran 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
32 |
31
|
expr |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑛 ∈ ω ) → ( 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) → ran 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ) |
33 |
32
|
rexlimdva |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( ∃ 𝑛 ∈ ω 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) → ran 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ) |
34 |
13 33
|
syl5bi |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) → ran 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ) |
35 |
34
|
imp |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) → ran 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
36 |
35
|
fmpttd |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) : ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ⟶ ( 𝒫 𝐴 ∩ Fin ) ) |
37 |
36
|
ffnd |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) Fn ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
38 |
36
|
frnd |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ran ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) ⊆ ( 𝒫 𝐴 ∩ Fin ) ) |
39 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
40 |
39
|
elin2d |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑦 ∈ Fin ) |
41 |
|
isfi |
⊢ ( 𝑦 ∈ Fin ↔ ∃ 𝑚 ∈ ω 𝑦 ≈ 𝑚 ) |
42 |
40 41
|
sylib |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∃ 𝑚 ∈ ω 𝑦 ≈ 𝑚 ) |
43 |
|
ensym |
⊢ ( 𝑦 ≈ 𝑚 → 𝑚 ≈ 𝑦 ) |
44 |
|
bren |
⊢ ( 𝑚 ≈ 𝑦 ↔ ∃ 𝑥 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) |
45 |
43 44
|
sylib |
⊢ ( 𝑦 ≈ 𝑚 → ∃ 𝑥 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) |
46 |
|
simprl |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑚 ∈ ω ) |
47 |
|
f1of |
⊢ ( 𝑥 : 𝑚 –1-1-onto→ 𝑦 → 𝑥 : 𝑚 ⟶ 𝑦 ) |
48 |
47
|
ad2antll |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑥 : 𝑚 ⟶ 𝑦 ) |
49 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
50 |
49
|
elin1d |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑦 ∈ 𝒫 𝐴 ) |
51 |
50
|
elpwid |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑦 ⊆ 𝐴 ) |
52 |
48 51
|
fssd |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑥 : 𝑚 ⟶ 𝐴 ) |
53 |
|
simplll |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝐴 ∈ dom card ) |
54 |
|
vex |
⊢ 𝑚 ∈ V |
55 |
|
elmapg |
⊢ ( ( 𝐴 ∈ dom card ∧ 𝑚 ∈ V ) → ( 𝑥 ∈ ( 𝐴 ↑m 𝑚 ) ↔ 𝑥 : 𝑚 ⟶ 𝐴 ) ) |
56 |
53 54 55
|
sylancl |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → ( 𝑥 ∈ ( 𝐴 ↑m 𝑚 ) ↔ 𝑥 : 𝑚 ⟶ 𝐴 ) ) |
57 |
52 56
|
mpbird |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑥 ∈ ( 𝐴 ↑m 𝑚 ) ) |
58 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝐴 ↑m 𝑛 ) = ( 𝐴 ↑m 𝑚 ) ) |
59 |
58
|
eleq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ↔ 𝑥 ∈ ( 𝐴 ↑m 𝑚 ) ) ) |
60 |
59
|
rspcev |
⊢ ( ( 𝑚 ∈ ω ∧ 𝑥 ∈ ( 𝐴 ↑m 𝑚 ) ) → ∃ 𝑛 ∈ ω 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) |
61 |
46 57 60
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → ∃ 𝑛 ∈ ω 𝑥 ∈ ( 𝐴 ↑m 𝑛 ) ) |
62 |
61 13
|
sylibr |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
63 |
|
f1ofo |
⊢ ( 𝑥 : 𝑚 –1-1-onto→ 𝑦 → 𝑥 : 𝑚 –onto→ 𝑦 ) |
64 |
63
|
ad2antll |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑥 : 𝑚 –onto→ 𝑦 ) |
65 |
|
forn |
⊢ ( 𝑥 : 𝑚 –onto→ 𝑦 → ran 𝑥 = 𝑦 ) |
66 |
64 65
|
syl |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → ran 𝑥 = 𝑦 ) |
67 |
66
|
eqcomd |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → 𝑦 = ran 𝑥 ) |
68 |
62 67
|
jca |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ ( 𝑚 ∈ ω ∧ 𝑥 : 𝑚 –1-1-onto→ 𝑦 ) ) → ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) |
69 |
68
|
expr |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑚 ∈ ω ) → ( 𝑥 : 𝑚 –1-1-onto→ 𝑦 → ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) ) |
70 |
69
|
eximdv |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑚 ∈ ω ) → ( ∃ 𝑥 𝑥 : 𝑚 –1-1-onto→ 𝑦 → ∃ 𝑥 ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) ) |
71 |
45 70
|
syl5 |
⊢ ( ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) ∧ 𝑚 ∈ ω ) → ( 𝑦 ≈ 𝑚 → ∃ 𝑥 ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) ) |
72 |
71
|
rexlimdva |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ∃ 𝑚 ∈ ω 𝑦 ≈ 𝑚 → ∃ 𝑥 ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) ) |
73 |
42 72
|
mpd |
⊢ ( ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) ∧ 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ∃ 𝑥 ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) |
74 |
73
|
ex |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) → ∃ 𝑥 ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) ) |
75 |
|
eqid |
⊢ ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) = ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) |
76 |
75
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) ↔ ∃ 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) 𝑦 = ran 𝑥 ) ) |
77 |
76
|
elv |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) ↔ ∃ 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) 𝑦 = ran 𝑥 ) |
78 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) 𝑦 = ran 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) |
79 |
77 78
|
bitri |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) ↔ ∃ 𝑥 ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ 𝑦 = ran 𝑥 ) ) |
80 |
74 79
|
syl6ibr |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) → 𝑦 ∈ ran ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) ) ) |
81 |
80
|
ssrdv |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝒫 𝐴 ∩ Fin ) ⊆ ran ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) ) |
82 |
38 81
|
eqssd |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ran ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) = ( 𝒫 𝐴 ∩ Fin ) ) |
83 |
|
df-fo |
⊢ ( ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) : ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) –onto→ ( 𝒫 𝐴 ∩ Fin ) ↔ ( ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) Fn ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ ran ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) = ( 𝒫 𝐴 ∩ Fin ) ) ) |
84 |
37 82 83
|
sylanbrc |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) : ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) –onto→ ( 𝒫 𝐴 ∩ Fin ) ) |
85 |
|
fodomnum |
⊢ ( ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∈ dom card → ( ( 𝑥 ∈ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↦ ran 𝑥 ) : ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) –onto→ ( 𝒫 𝐴 ∩ Fin ) → ( 𝒫 𝐴 ∩ Fin ) ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) ) |
86 |
12 84 85
|
sylc |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝒫 𝐴 ∩ Fin ) ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
87 |
|
domtr |
⊢ ( ( ( 𝒫 𝐴 ∩ Fin ) ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ∧ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ≼ 𝐴 ) → ( 𝒫 𝐴 ∩ Fin ) ≼ 𝐴 ) |
88 |
86 10 87
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝒫 𝐴 ∩ Fin ) ≼ 𝐴 ) |
89 |
|
pwexg |
⊢ ( 𝐴 ∈ dom card → 𝒫 𝐴 ∈ V ) |
90 |
89
|
adantr |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → 𝒫 𝐴 ∈ V ) |
91 |
|
inex1g |
⊢ ( 𝒫 𝐴 ∈ V → ( 𝒫 𝐴 ∩ Fin ) ∈ V ) |
92 |
90 91
|
syl |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝒫 𝐴 ∩ Fin ) ∈ V ) |
93 |
|
infpwfidom |
⊢ ( ( 𝒫 𝐴 ∩ Fin ) ∈ V → 𝐴 ≼ ( 𝒫 𝐴 ∩ Fin ) ) |
94 |
92 93
|
syl |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → 𝐴 ≼ ( 𝒫 𝐴 ∩ Fin ) ) |
95 |
|
sbth |
⊢ ( ( ( 𝒫 𝐴 ∩ Fin ) ≼ 𝐴 ∧ 𝐴 ≼ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝒫 𝐴 ∩ Fin ) ≈ 𝐴 ) |
96 |
88 94 95
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom card ∧ ω ≼ 𝐴 ) → ( 𝒫 𝐴 ∩ Fin ) ≈ 𝐴 ) |